$SU(2)^2$-invariant $G_2$-instantons

TL;DR

This paper systematically studies $G_2$-instantons with $SU(2)^2$ symmetry, providing foundational theory, existence and classification results, and analyzing their behavior on specific $G_2$ manifolds with different volume growths.

Contribution

It develops the foundational theory for $SU(2)^2$-invariant $G_2$-instantons and presents new existence, non-existence, and classification results, including explicit examples and phenomena like bubbling.

Findings

Existence and classification results for $G_2$-instantons with $SU(2)^2$ symmetry.

Explicit examples demonstrating bubbling and removable singularities.

Analysis of instantons on $ ext{R}^4 imes S^3$ with different volume growths.

Abstract

We initiate the systematic study of $G_2$-instantons with $SU(2)^2$-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on $\mathbb{R}^4\times S^3$ with its two explicitly known distinct holonomy $G_2$ metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of $G_2$-instantons where "bubbling" and "removable singularity" phenomena occur in the limit.